I know this is an old thread but for anyone in the same situation then post brexit this is the starting point: https://www.gov.uk/goods-sent-from-abroad Note that "excise goods" basically means alcohol and tobacco.

Basically if the value of the goods is under £135 (roughly $160) then the seller in the US will have to pay the VAT (20%) to the UK government, they have to be registered with UK Gov to do this. If over then the buyer has to pay the courier the VAT and any duty (they then pass this on to the government). At the UK end most couriers have a standard "release fee" that they charge on top of any VAT or duty payable, some want the payment up-front, others bill you later.

The rules are essentially the same for sending stuff to the EU as the procedures were sorted out whilst we were still a member.